Estimation of Received Signals at Arbitrary

Remote Locations based on Estimation of Arriving

Waves by Compressed Sensing

Tomoya Sugimoto, Hisato Iwai, and Hideichi Sasaoka Graduate School of Science of Engineering, Doshisha University

1-3, Tatara Miyako-dani, Kyotanabe, Kyoto, 610-0394 Japan

Abstract – Propagation characteristics have locality in a

multipath fading environment. The locality is utilized for various techniques using radio wave. Recently, on the other hand, new techniques to break the locality have been studied. In the

techniques, received signals at arbitrary locations are estimated from remote receiving points. We present in this paper a new estimation technique of received signals utilizing Compressed

Sensing (CS). In the technique, we estimate directions of arrival and each complex amplitude of arriving waves by CS, and calculate a received signal at a remote location based on the

estimated results. We present the technique and evaluate the estimation performance through computer simulations.

Index Terms — Compressed sensing, Estimation of received

signal, Multipath environment.

1. Introduction

Propagation characteristics of radio waves are local in a

multipath fading environment. Various techniques based on

the characteristics have been developed. An example is found

in wireless physical layer security [1]. The technique utilizes

the propagation characteristics of wireless channels between

legitimate two users in order to transmit messages securely. In

the case that we are remote from the users, we cannot

eavesdrop on the messages because we do not know the

legitimate propagation characteristics.

On the other hand, new techniques to break the locality have

been studied in recent years. If we can estimate the received

signals at the legitimate destinations, the possibility of

eavesdropping in the physical layer security increases. Thus

the estimation techniques of received signals at remote

locations can be applied to various radio techniques.

In this paper, we regard complex amplitude at a remote

location as the received signal. In the previous study [2], we

estimate the complex amplitude at the remote location based

on the directions of arrival (DOAs) and each complex

amplitude of the arriving waves. We estimate the DOAs by

the Multiple Signal Classification (MUSIC) method, and the

complex amplitude by the least squares method. In this paper,

we estimate DOAs and complex amplitude simultaneously by

Compressed Sensing (CS). CS is a technique to obtain a

solution from an underdetermined linear system [3, 4]. Based

on the DOAs and each complex amplitude of arriving waves,

we calculate a received signal at a remote location. We

evaluate the estimation performance via computer simulations.

2. Estimation of Received Signal at an Arbitrary

Remote Location by Compressed Sensing

CS is a technique to provide an optimum solution of an

underdetermined system taking advantage of the prior

knowledge that the true solution is sparse. We consider the

linear system having M equations and N unknowns as

y = Ax + n (1)

where A ϵ M×N is a sensing matrix, n ϵ M is a noise vector,

x ϵ N is an unknown vector, and y ϵ M is a measurement

vector. In the case of M<N, we cannot obtain the true solution

generally. However, in the case that the true solution is a

sparse vector, in other words, the number of the non-zero

elements of x is less than that of the equations, we can

calculate the optimum solution. This is the basic idea to obtain

the optimum solution by CS. CS is utilized for various fields

including communications [5, 6].

Figure 1. Assumed environment and estimation model.

Figure 1 shows the assumed environment and the estimation

system used in the estimation of the received signals.

Assuming a two-dimensional space, an M-elements circular

array is placed, and the radius of the element arrangement is

R. In the polar coordinate, the origin of the coordinate is set at

the center of the circle. The position of the target point where

the received signal is estimated is denoted (r, φ). The whole

angular range of the estimation, -180°~180° in this case, is

divided into N small angular bins. θn is the angle of the n-th

angular bin. xn is the n-th element of the vector x. It

corresponds to the complex amplitude of the arriving wave

contained within the n-th angular bin. ym

is the m-th element

12

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Arrivingwave

R

Target point(r , φ)

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Antennaelement

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Proceedings of ISAP2016, Okinawa, Japan

Copyright ©2016 by IEICE

4C3-4

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of the vector y. It is the received signal at the m-th antenna

element. We can calculate the optimum solution x by CS,

setting the estimation system as the number of the angular

division N is much larger than that of the arriving waves.

In this paper, we estimate the received signal s at the target

point considering the phase difference based on the distance

from the origin to the target point by the following formula

s = ∑ xn

N

n=1exp {-j2π

r

λcos (θn-φ)} (2)

where λ is the wavelength of the carrier frequency of the radio

waves. In the above estimation formula, all arriving waves are

assumed plane waves.

3. Evaluation of Estimation Performance

We evaluate estimation performance of received signals at

an arbitrary remote location by CS via computer simulations.

Table 1 summarizes the assumed values of the multipath

environment and the estimation system.

Figure 2 shows the amplitude ratio of the estimated

received signal to the actual at the two target points

( (r, φ)=(10λ, 10°) and (100λ, 10°) ). In the figure, 100

independent trials are carried out varying the DOAs and the

phases of the arriving waves randomly. In the case of r=10λ,

the errors of the estimation are very small. On the other hand

when r=100λ, the errors are large. Here we use 90% value of

the distribution of the amplitude ratio as a measure of the

estimation accuracy. The 90% value is a value of the

amplitude ratio where ninety percent of the absolute values of

the amplitude ratios of the independent trials are below the

value. In the case of r=100λ, the 90% value is 2.2 dB.

Figure 3 shows the 90% value over the variation of the

distance to the target points. We calculate the received signals

with both the estimated DOAs and the actual. We add certain

intentional angular error ψ to each actual DOA in order to

consider the effect of the DOA estimation error. In the case of

ψ=0°, the 90% values are very small. It appears that we can

obtain complex amplitude of arriving waves correctly by CS.

The results shown in the figure shows the errors of the

estimated DOAs cause the degradation of the estimation of the

received signals. It is shown that the 90% values obtained

using the estimated DOAs are close to that with 0.02°

intentional error. It indicates that the angular errors of the

estimation are about 0.02°. Since DOA is discretely expressed

with 0.01° separation in the estimation, it inherently includes

the quantization error up to 0.005°. However the above result

shows the DOA estimation error is larger than the quantization

error. Therefore, we see that the improvement of the DOA

estimation accuracy is a key in order to improve the whole

estimation performance.

4. Conclusion

In this paper, we present a technique to estimate received

signals at an arbitrary remote location based on estimation of

arriving waves by CS. We show that the errors of the DOA

estimation deteriorate the estimation. It is our future task to

improve the DOA estimation accuracy to improve the whole

estimation performance.

Figure 2. Example of estimation using CS.

Figure 3. Influence of distance to target points.

Table 1. Values of environment and estimation system.

Multipath environment

Number of arriving

waves 3

Arriving angle Random (-180°~180°)

Complex amplitude of

arriving waves

Amplitude : Unity

Phase : Random (-180°~180°)

SNR 40 dB

Estimation system

Antenna separation λ

Number of antennas 20

Number of snapshots 100

Bin size 0.01°

References

[1] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays,” IEEE Trans. Signal Processing, vol. 58, no. 3, pp. 1875-1888, March 2010.

[2] M. Tanaka, H. Iwai, and H. Sasaoka, “Estimation of received signal at an arbitrary remote location using MUSIC method,” IEICE Trans. Commun., vol. E98-B, no. 5, pp. 806-813, May 2015.

[3] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289-1306, April 2006.

[4] E. J. Candes, M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag., vol. 25, no. 2, pp. 21-30, March 2008.

[5] K. Hayashi, M. Nagahara, and T. Tanaka, “A user’s guide to compressed sensing for communications systems,” IEICE Trans. Commun., vol. E96-B, no. 3, pp. 685-712, March 2013

[6] T. Terada, T. Nishimura, Y. Ogawa, T. Ohgane, and H. Yamada, “DOA estimation for multi-band signal sources using compressed sensing techniques with Khatri-rao processing,” IEICE Trans. Commun., vol. E97-B, no. 10, pp. 2110-2117, Oct. 2014

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